Subgroups — #10621615

Order 672.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41⋊ΓL(1,𝔽₈)	C₂	1	3	yes	maximal, complemented
2	C₇×H(5,𝔽₂)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇⋊SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
4	Q₈×ΓL(1,𝔽₈)	—	4	1	no	maximal
5	C₇×#41	—	3	1	no	—
6	C₇×#41	C₂×C₃	1	3	yes	—
7	C₂×C₇×D₄	—	3	3	no	—
8	#994	—	7	1	no	maximal, Hall
9	C₄×ΓL(1,𝔽₈)	—	4	3	no	—
10	C₂³×C₇	—	3	2	no	—
11	C₇×Q₈	A₄	1	1	yes	characteristic
12	C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
13	C₇×D₄	—	3	3	no	—
14	C₇×D₄	—	3	3	no	—
15	C₂×C₄×C₇	—	3	3	no	—
16	#41⋊C₃	—	7	3	no	—
17	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
18	H(5,𝔽₂)	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
19	C₂²×C₇	—	3	3	no	—
20	C₂²×C₇	—	12	2	no	—
21	C₄×C₇	—	3	1	no	—
22	C₄×C₇	C₂×A₄	1	3	yes	—
23	C₃×Q₈	—	28	1	no	—
24	SL(2,𝔽₃)	—	7	1	no	—
25	ΓL(1,𝔽₈)	—	4	1	no	Hall
26	#41	—	3	1	no	—
27	#41	C₂×ΓL(1,𝔽₈)	1	3	yes	—
28	C₂×D₄	—	3	3	no	—
29	C₂×C₇	—	6	3	no	—
30	C₂×C₇	C₂²×A₄	1	1	yes	characteristic
31	C₃×C₄	—	28	3	no	—
32	C₂³	—	3	2	no	—
33	Q₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
34	Q₈	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
35	D₄	—	3	3	no	—
36	D₄	—	3	3	no	—
37	C₂×C₄	—	3	3	no	—
38	C₇	#994	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₃	—	28	1	no	—
40	C₂²	—	3	3	no	—
41	C₂²	—	12	2	no	—
42	C₄	—	3	1	no	—
43	C₄	C₂×C₂²⋊ΓL(1,𝔽₈)	1	3	yes	—
44	C₃	—	28	1	no	Sylow
45	C₂	—	6	3	no	—
46	C₂	C₂²×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇×Q₈	C₂²×C₃
Fitting	C₇×H(5,𝔽₂)	C₃
socle	C₂×C₇	C₂²×A₄
hypercenter	Q₈	C₂²⋊ΓL(1,𝔽₈)
LCS	G ▷ C₇×Q₈
UCS	C₁ ◁ C₂ ◁ Q₈

Splittings G = N ⋊ H with N normal; H ≅ G/N.

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.